Theorem prnz 4730

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4715	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4294	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436  ∅c0 4283  {cpr 4578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3905  df-un 3907  df-nul 4284  df-sn 4577  df-pr 4579

This theorem is referenced by:  opnz  5413  propssopi  5448  fiint  9211  wilthlem2  27004  upgrbi  29069  wlkvtxiedg  29601  shincli  31337  chincli  31435  constrextdg2lem  33756  spr0nelg  47506  sprvalpwn0  47513